Why don't interfering radio stations both play at the same time?

Your mention of the frequency (97.5 MHz) tells us this is an FM receiver. (AM will behave differently, as will other modulation schemes).

Because FM is encoded by modulating the signal frequency, anything to do with AM is undesirable. To deal with this, most receivers over-amplify the signal until it becomes larger than the later stages can pass. The signal then "clips" to the voltage of that amplifier. This stage is called a "limiter"--it limits the amplitude to some fixed value. In theory, any signal weaker than that drops out and just becomes noise, and any signal stronger than that has a very nice fixed level that the FM detector can handle without having to worry about amplitude variations.

The amplifier-limiter stages create a phenomenon called "capture", where the strong signal tends to eliminate the weaker one. This is why you hear only one station.

If the signals were very close in strength, you would indeed hear them "mixing together", but that only happens for a fraction of a second as the signal levels rapidly change (presumably, you are in a vehicle), so you normally don't hear that.

Here's my take on it:

While the carrier signals may have the same frequency, they have different phases. As the PLL in the FM decoder drifts, it locks first onto one phase and then onto the other, leading to alternating broadcasts being decoded as time goes on.

FM radio signals are not Additive. If you add two FM signals together, you do not get one "mixed" FM signal. You just get noise.

Fortunately, FM receivers are good at picking an FM signal out of noise, so your radio is able to separate one - or the other - of the two signals out of the noise. It does this by locking onto the phase of one carrier signal, which allows it to treat the out-of-phase signal as amplitude noise, which it always rejects.

The problem is that, as the two signals drift in and out of phase (something that happens because of the music, even if the two transmitters are perfectly phase locked, which they aren't), the locked signal gets captured by one - or the other - of the two transmitters. Which signal gets followed depends not on the amplitude (unless the amplitude difference is big enough), but on (1) the music (which is doing transient drifts to the carrier phase) and (2) the transmitter phase drift, and (3) the receiver (which will have a drift preference at any moment).

It is possible to make FM receivers which are not phase-locked. You can, for example, convert the frequency into amplitude before doing conversion. In that case, your mixed FM signal wouldn't work at all, because the noise wouldn't have a recoverable carrier frequency.